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  <section id="numpy-random-generator-negative-binomial">
<h1>numpy.random.Generator.negative_binomial<a class="headerlink" href="#numpy-random-generator-negative-binomial" title="Link to this heading">#</a></h1>
<p>method</p>
<dl class="py method">
<dt class="sig sig-object py" id="numpy.random.Generator.negative_binomial">
<span class="sig-prename descclassname"><span class="pre">random.Generator.</span></span><span class="sig-name descname"><span class="pre">negative_binomial</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.Generator.negative_binomial" title="Link to this definition">#</a></dt>
<dd><p>Draw samples from a negative binomial distribution.</p>
<p>Samples are drawn from a negative binomial distribution with specified
parameters, <em class="xref py py-obj">n</em> successes and <em class="xref py py-obj">p</em> probability of success where <em class="xref py py-obj">n</em>
is &gt; 0 and <em class="xref py py-obj">p</em> is in the interval (0, 1].</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>n</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Parameter of the distribution, &gt; 0.</p>
</dd>
<dt><strong>p</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Parameter of the distribution. Must satisfy 0 &lt; p &lt;= 1.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape.  If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n,</span> <span class="pre">k)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn.  If size is <code class="docutils literal notranslate"><span class="pre">None</span></code> (default),
a single value is returned if <code class="docutils literal notranslate"><span class="pre">n</span></code> and <code class="docutils literal notranslate"><span class="pre">p</span></code> are both scalars.
Otherwise, <code class="docutils literal notranslate"><span class="pre">np.broadcast(n,</span> <span class="pre">p).size</span></code> samples are drawn.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>Drawn samples from the parameterized negative binomial distribution,
where each sample is equal to N, the number of failures that
occurred before a total of n successes was reached.</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>The probability mass function of the negative binomial distribution is</p>
<div class="math notranslate nohighlight">
\[P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},\]</div>
<p>where <span class="math notranslate nohighlight">\(n\)</span> is the number of successes, <span class="math notranslate nohighlight">\(p\)</span> is the
probability of success, <span class="math notranslate nohighlight">\(N+n\)</span> is the number of trials, and
<span class="math notranslate nohighlight">\(\Gamma\)</span> is the gamma function. When <span class="math notranslate nohighlight">\(n\)</span> is an integer,
<span class="math notranslate nohighlight">\(\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}\)</span>, which is
the more common form of this term in the pmf. The negative
binomial distribution gives the probability of N failures given n
successes, with a success on the last trial.</p>
<p>If one throws a die repeatedly until the third time a “1” appears,
then the probability distribution of the number of non-“1”s that
appear before the third “1” is a negative binomial distribution.</p>
<p>Because this method internally calls <code class="docutils literal notranslate"><span class="pre">Generator.poisson</span></code> with an
intermediate random value, a ValueError is raised when the choice of 
<span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(p\)</span> would result in the mean + 10 sigma of the sampled
intermediate distribution exceeding the max acceptable value of the 
<code class="docutils literal notranslate"><span class="pre">Generator.poisson</span></code> method. This happens when <span class="math notranslate nohighlight">\(p\)</span> is too low 
(a lot of failures happen for every success) and <span class="math notranslate nohighlight">\(n\)</span> is too big (
a lot of successes are allowed).
Therefore, the <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(p\)</span> values must satisfy the constraint:</p>
<div class="math notranslate nohighlight">
\[n\frac{1-p}{p}+10n\sqrt{n}\frac{1-p}{p}&lt;2^{63}-1-10\sqrt{2^{63}-1},\]</div>
<p>Where the left side of the equation is the derived mean + 10 sigma of
a sample from the gamma distribution internally used as the <span class="math notranslate nohighlight">\(lam\)</span>
parameter of a poisson sample, and the right side of the equation is
the constraint for maximum value of <span class="math notranslate nohighlight">\(lam\)</span> in <code class="docutils literal notranslate"><span class="pre">Generator.poisson</span></code>.</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="r30b32598df4a-1" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>Weisstein, Eric W. “Negative Binomial Distribution.” From
MathWorld–A Wolfram Web Resource.
<a class="reference external" href="https://mathworld.wolfram.com/NegativeBinomialDistribution.html">https://mathworld.wolfram.com/NegativeBinomialDistribution.html</a></p>
</div>
<div class="citation" id="r30b32598df4a-2" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<p>Wikipedia, “Negative binomial distribution”,
<a class="reference external" href="https://en.wikipedia.org/wiki/Negative_binomial_distribution">https://en.wikipedia.org/wiki/Negative_binomial_distribution</a></p>
</div>
</div>
<p class="rubric">Examples</p>
<div class="try_examples_outer_container docutils container" id="ff527a33-5586-48c5-b7e9-b63bc5e2e082">
<div class="try_examples_button_container"><button class="try_examples_button" onclick="window.tryExamplesShowIframe('ff527a33-5586-48c5-b7e9-b63bc5e2e082','bdd8ffa2-6692-4e2b-86fa-9f2034116bfd','db00e891-30ef-4972-af87-2df587737543','../../../lite/tree/../notebooks/index.html?path=22989b01_b6b0_4250_8a38_08d32edeaca5.ipynb','None')">Try it in your browser!</button></div><div class="try_examples_content docutils container">
<p>Draw samples from the distribution:</p>
<p>A real world example. A company drills wild-cat oil
exploration wells, each with an estimated probability of
success of 0.1.  What is the probability of having one success
for each successive well, that is what is the probability of a
single success after drilling 5 wells, after 6 wells, etc.?</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">()</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">negative_binomial</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">100000</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">):</span> 
<span class="gp">... </span>   <span class="n">probability</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">s</span><span class="o">&lt;</span><span class="n">i</span><span class="p">)</span> <span class="o">/</span> <span class="mf">100000.</span>
<span class="gp">... </span>   <span class="nb">print</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="s2">&quot;wells drilled, probability of one success =&quot;</span><span class="p">,</span> <span class="n">probability</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div id="db00e891-30ef-4972-af87-2df587737543" class="try_examples_outer_iframe  hidden"><div class="try_examples_button_container"><button class="try_examples_button" onclick="window.tryExamplesHideIframe('ff527a33-5586-48c5-b7e9-b63bc5e2e082','db00e891-30ef-4972-af87-2df587737543')">Go Back</button><button class="try_examples_button" onclick="window.openInNewTab('ff527a33-5586-48c5-b7e9-b63bc5e2e082','db00e891-30ef-4972-af87-2df587737543')">Open In Tab</button></div><div id="bdd8ffa2-6692-4e2b-86fa-9f2034116bfd" class="jupyterlite_sphinx_iframe_container"></div></div><script>document.addEventListener("DOMContentLoaded", function() {window.loadTryExamplesConfig("../../../try_examples.json");});</script></dd></dl>

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